3.2.40 \(\int \frac {\sin ^2(\log (x))}{x} \, dx\) [140]

Optimal. Leaf size=17 \[ \frac {\log (x)}{2}-\frac {1}{2} \cos (\log (x)) \sin (\log (x)) \]

[Out]

1/2*ln(x)-1/2*cos(ln(x))*sin(ln(x))

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Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2715, 8} \begin {gather*} \frac {\log (x)}{2}-\frac {1}{2} \sin (\log (x)) \cos (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[Log[x]]^2/x,x]

[Out]

Log[x]/2 - (Cos[Log[x]]*Sin[Log[x]])/2

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rubi steps

\begin {align*} \int \frac {\sin ^2(\log (x))}{x} \, dx &=\text {Subst}\left (\int \sin ^2(x) \, dx,x,\log (x)\right )\\ &=-\frac {1}{2} \cos (\log (x)) \sin (\log (x))+\frac {1}{2} \text {Subst}(\int 1 \, dx,x,\log (x))\\ &=\frac {\log (x)}{2}-\frac {1}{2} \cos (\log (x)) \sin (\log (x))\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 16, normalized size = 0.94 \begin {gather*} \frac {\log (x)}{2}-\frac {1}{4} \sin (2 \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[Log[x]]^2/x,x]

[Out]

Log[x]/2 - Sin[2*Log[x]]/4

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Maple [A]
time = 0.06, size = 14, normalized size = 0.82

method	result	size
derivativedivides	\(\frac {\ln \left (x \right )}{2}-\frac {\cos \left (\ln \left (x \right )\right ) \sin \left (\ln \left (x \right )\right )}{2}\)	\(14\)
default	\(\frac {\ln \left (x \right )}{2}-\frac {\cos \left (\ln \left (x \right )\right ) \sin \left (\ln \left (x \right )\right )}{2}\)	\(14\)
risch	\(\frac {\ln \left (x \right )}{2}+\frac {i x^{2 i}}{8}-\frac {i x^{-2 i}}{8}\)	\(24\)
norman	\(\frac {\tan ^{3}\left (\frac {\ln \left (x \right )}{2}\right )+\frac {\ln \left (x \right )}{2}+\ln \left (x \right ) \left (\tan ^{2}\left (\frac {\ln \left (x \right )}{2}\right )\right )+\frac {\ln \left (x \right ) \left (\tan ^{4}\left (\frac {\ln \left (x \right )}{2}\right )\right )}{2}-\tan \left (\frac {\ln \left (x \right )}{2}\right )}{\left (1+\tan ^{2}\left (\frac {\ln \left (x \right )}{2}\right )\right )^{2}}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(ln(x))^2/x,x,method=_RETURNVERBOSE)

[Out]

1/2*ln(x)-1/2*cos(ln(x))*sin(ln(x))

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Maxima [A]
time = 0.27, size = 12, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \log \left (x\right ) - \frac {1}{4} \, \sin \left (2 \, \log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(log(x))^2/x,x, algorithm="maxima")

[Out]

1/2*log(x) - 1/4*sin(2*log(x))

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Fricas [A]
time = 0.36, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{2} \, \cos \left (\log \left (x\right )\right ) \sin \left (\log \left (x\right )\right ) + \frac {1}{2} \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(log(x))^2/x,x, algorithm="fricas")

[Out]

-1/2*cos(log(x))*sin(log(x)) + 1/2*log(x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (15) = 30\).
time = 1.70, size = 156, normalized size = 9.18 \begin {gather*} \frac {\log {\left (x \right )} \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )}}{2 \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 4 \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 2} + \frac {2 \log {\left (x \right )} \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )}}{2 \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 4 \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 2} + \frac {\log {\left (x \right )}}{2 \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 4 \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 2} + \frac {2 \tan ^{3}{\left (\frac {\log {\left (x \right )}}{2} \right )}}{2 \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 4 \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 2} - \frac {2 \tan {\left (\frac {\log {\left (x \right )}}{2} \right )}}{2 \tan ^{4}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 4 \tan ^{2}{\left (\frac {\log {\left (x \right )}}{2} \right )} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(ln(x))**2/x,x)

[Out]

log(x)*tan(log(x)/2)**4/(2*tan(log(x)/2)**4 + 4*tan(log(x)/2)**2 + 2) + 2*log(x)*tan(log(x)/2)**2/(2*tan(log(x
)/2)**4 + 4*tan(log(x)/2)**2 + 2) + log(x)/(2*tan(log(x)/2)**4 + 4*tan(log(x)/2)**2 + 2) + 2*tan(log(x)/2)**3/
(2*tan(log(x)/2)**4 + 4*tan(log(x)/2)**2 + 2) - 2*tan(log(x)/2)/(2*tan(log(x)/2)**4 + 4*tan(log(x)/2)**2 + 2)

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Giac [A]
time = 7.75, size = 12, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \log \left (x\right ) - \frac {1}{4} \, \sin \left (2 \, \log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(log(x))^2/x,x, algorithm="giac")

[Out]

1/2*log(x) - 1/4*sin(2*log(x))

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Mupad [B]
time = 0.38, size = 12, normalized size = 0.71 \begin {gather*} \frac {\ln \left (x\right )}{2}-\frac {\sin \left (2\,\ln \left (x\right )\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(log(x))^2/x,x)

[Out]

log(x)/2 - sin(2*log(x))/4

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